CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-31

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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For unrestarted GMRES, the residual norms are nonincreasing

Statement

Let $x_{m}$ be an unrestarted GMRES iterate at step $m$ . Then

$∥ b - A x_{m + 1} ∥_{2} \leq ∥ b - A x_{m} ∥_{2} (m \geq 0) .$

Facts & Assumptions

Given: Unrestarted GMRES iterates $x_{m}$ and $x_{m + 1}$ for the same initial guess $x_{0}$ .

[L1]

Each GMRES iterate minimizes the residual norm over its affine Krylov space $x_{0} + K_{m} (A, r_{0})$ (GMRES minimizes the residual norm over the affine Krylov space $x_{0} + K_{m} (A, r_{0})$ ).

Proof

technique · direct

1.1

Because $K_{m} (A, r_{0}) \subseteq K_{m + 1} (A, r_{0})$ , one has $x_{0} + K_{m} (A, r_{0}) \subseteq x_{0} + K_{m + 1} (A, r_{0})$ . In particular, $x_{m}$ is an admissible competitor for the $(m + 1)$ st minimization problem.

givenalgebra

2.1

By [L1], $x_{m + 1}$ minimizes the residual norm over the larger affine space, so $∥ b - A x_{m + 1} ∥_{2} \leq ∥ b - A x_{m} ∥_{2} .$

L1step 1.1∎

Depends on

GMRES minimizes the residual norm over the affine Krylov space $x_0+K_m(A,r_0)$

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Lloyd N. Trefethen and David Bau III, Numerical Linear Algebra (standard reference, not scraped)